In `triangleABC` (not shown), `AC bot BC ad cos angleABC=(12)/(13)`. What is the value of `tan angleABC`?

Options

A. `(5)/(13)`
B. `(5)/(12)`
C. `(12)/(13)`
D. `(12)/(5)`

Answer

To find the value of \( \tan \angle ABC \) in triangle \( ABC \) where \( AC \) is perpendicular to \( BC \) and \( \cos \angle ABC = \frac{12}{13} \), we can follow these steps:

### Step 1: Understand the relationship of cosine in a right triangle
In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse. Here, \( \cos \angle ABC = \frac{BC}{AB} \).

### Step 2: Assign values based on cosine
From the given \( \cos \angle ABC = \frac{12}{13} \), we can assign:
- \( BC = 12 \) (adjacent side)
- \( AB = 13 \) (hypotenuse)

### Step 3: Use the Pythagorean theorem to find the opposite side
We need to find the length of the opposite side \( AC \). According to the Pythagorean theorem:
\[ AB^2 = BC^2 + AC^2 \]
Substituting the known values:
\[ 13^2 = 12^2 + AC^2 \]
Calculating the squares:
\[ 169 = 144 + AC^2 \]
Now, isolate \( AC^2 \):
\[ AC^2 = 169 - 144 \]
\[ AC^2 = 25 \]
Taking the square root gives:
\[ AC = 5 \]

### Step 4: Calculate \( \tan \angle ABC \)
Now that we have \( AC \) (the opposite side) and \( BC \) (the adjacent side), we can find \( \tan \angle ABC \):
\[ \tan \angle ABC = \frac{AC}{BC} = \frac{5}{12} \]

### Final Answer
Thus, the value of \( \tan \angle ABC \) is \( \frac{5}{12} \).

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